EAST RUTHERFORD, N.J. — The showboating Yankees may win more often, but the staid New York Football Giants have quietly produced consistent teams and a handful of championships, staking a worthy claim as the city’s most revered franchise. Owned by the same family since its founding in 1925 as one of the N.F.L.’s earliest members, the team is like a regal trans-Atlantic ocean liner that rarely strays off course.

Until, that is, this year.

The Giants, who in 2012 won the last championship for a major professional sports franchise in the New York area, have ignominiously run aground with a 1-5 record and look far from ending that six-year title drought.

Fans now loudly accuse the Giants’ brain trust of not only bungling this season but ruining the team’s future, too.

At the center of it all is Eli Manning, the beloved quarterback who brought the team its last two championships, winning the Most Valuable Player Award in each Super Bowl, but who is well past his prime.

The team’s management is now intensifying an exhaustive overhaul that began last season, which the team finished 3-13, an effort that could lead to seismic change and the end of Manning’s storied tenure.

For now, Manning has the support of the team’s front office and coaches, but the Giants will probably acquire another quarterback next year through the draft or in free agency, according to people familiar with the team’s recent deliberations. They spoke on the condition of anonymity because the Giants permit only a select few to speak publicly about team matters.

A new quarterback could compete for the starting job or even lead the Giants to cut Manning, who will turn 38 in January and who would cost the team $23.2 million against the salary cap next season.

It would only continue a radical rebuilding: Two-thirds of the roster from last season has been jettisoned. How well Manning, the starter since 2004, plays in the remaining 10 games this season will also play a part in determining his fate.

If the Giants move on from Manning next season, many will no doubt ask why the team did not use its No. 2 overall draft choice in April to take a quarterback. That class was rich with quarterback talent, including Sam Darnold, the Southern California quarterback selected at No. 3 by the Jets.

The Giants instead chose running back Saquon Barkley, who has had a spectacular rookie season. They picked Barkley because the team’s evaluators had unanimously ranked him above any other player in the draft, according to people with knowledge of the team’s thinking, and not because anyone in the Giants’ hierarchy — including the chief executive and co-owner John Mara — had insisted that Manning remain as quarterback for this season or in 2019. No quarterback in the draft was, in their view, more valuable than Barkley at the time.

The Giants also believed Manning could still contribute capably and rebound from a subpar season in 2017.

With Darnold in uniform and bringing new life to the Jets, however, that single decision by the Giants could define both franchises for many years to come.

For now, the Giants must deal with a fan base fuming over a 1-5 record, the same tally after six games a year ago. They are in last place in the N.F.C. East.

Heading into their game Monday night in Atlanta, the Giants had lost 18 of their last 22 games and have had two winning seasons since their Super Bowl win after the 2011 season.

“I’m sick about it, particularly the way we’ve disappointed our fans,” Mara, whose family shares ownership with the Tisch family, told reporters last week at the N.F.L. owners’ meetings in New York.

He added: “I don’t think you ever expect to go 1-5 — two years in a row. I’m still embarrassed.”

Players’ frustrations and anger have spilled into the open.

Odell Beckham Jr., one of the Giants’ brightest young stars, questioned the team’s energy, heart and play-calling in an ESPN interview this month. Given a chance to endorse Manning, Beckham passed; on Friday, though, Beckham praised Manning for his preparation and leadership.

Still, inside the Giants’ locker room, there has been a concerted effort to foster unity, including a players-only meeting in recent weeks.

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Manning fumbled during a game against the Cowboys last month. The Giants are 1-5 and in the middle of a franchise overhaul.

CreditTim Heitman/USA Today Sports, via Reuters

Wide receiver Sterling Shepard sought to counter the impression made in some news reports that players had had it with Manning.

“As far as in the locker room, we have total faith in him,” Shepard said. “We picked him as our starting quarterback at the beginning of the season, and we’re going to ride with that.”

Others see Beckham as a lightning rod. But that does not mean he is being dishonest about the team’s struggles. A former Giant from last season said via text message: “Odell is visibly upset, and rightfully so. He just signed a big deal, and he wants to go out and prove he deserves it, and no matter whose fault it is, it’s still not happening right now.”

Manning, who is set to make his 221st start on Monday, for years has typified the poised Giants mien. But lately, he has borne the brunt of fans’ wrath.

“We still believe in him,” Mara said of Manning on Wednesday. “I know he’s the punching bag right now, but a lot of guys need to play better right now. It’s not just him.”

Nonetheless, Manning’s Giants career appears to be nearing an end.

Standing at his locker on Thursday, Manning said he was working hard to tune out the negative vibe enveloping the team.

“That’s always a challenge,” he said. “It’s just a challenge to try to focus on the things that you can control, and that’s us and your assignments, the game, going out there and playing well. Keep the focus on that, and we’ll be O.K.”

Manning’s backups, Alex Tanney and the rookie Kyle Lauletta, have little experience; Tanney, a 30-year-old journeyman who is No. 2 on the team’s depth chart, is best known for trick-shot videos he made in college.

It’s hard to recall that just several months ago, Manning was still seen as the team’s cornerstone.

When Ben McAdoo, the Giants’ coach for most of last season, wanted Manning temporarily benched during the latter stages of a dismal season so he could try out other quarterbacks on the roster, furious fans considered the move a slap in the face to Manning.

Many believe that episode led to the Giants’ not drafting a quarterback, and also influenced their decision to replace McAdoo with the former Minnesota offensive coordinator Pat Shurmur and to name the former Carolina Panthers general manager Dave Gettleman as Reese’s successor.

Boomer Esiason, a former N.F.L. quarterback who is now a CBS analyst, insisted that the Giants lost out on two potential coaching candidates, Matt Patricia and Josh McDaniels, because of their commitment to Manning.

“Jerry Reese lost his job. Tom Coughlin lost his job. Ben McAdoo lost his job. Ereck Flowers lost his job,” Esiason said in a telephone interview. “There’s one constant, and that happens to be the $22-million-per-year quarterback.”

But Esiason, like many observers, also recognizes the significant erosion of talent around Manning.

Reese, who started as the team’s general manager in 2007, presided over two Super Bowl-winning teams and was responsible for drafting standouts like Beckham, defensive end Jason Pierre-Paul and safety Landon Collins. But an examination of dozens of Reese’s draft picks across many years reveals a recurring failure to find players with N.F.L.-level talent.

Since 2011, 25 of Reese’s picks — more than half of his selections — are no longer active on any N.F.L. roster, and only nine are still with the Giants. That does not include a single offensive lineman, a position Reese failed to fortify again and again.

When Gettleman arrived, he quickly engineered a sweeping roster makeover. As of this week, only 17 Giants players had been on the team a year ago. Even a day after training camp ended, when the roster was trimmed to the mandatory 53 players, the Giants put in a league-high six additional waiver claims. It was an unmistakable rebuke of the talent that Reese had assembled.

Beginning last season, a porous offensive line has routinely been blamed for Manning’s skittishness in the pocket and his ineffectiveness throwing the football downfield. Gettleman invested heavily in retooling the line this off-season, signing Nate Solder and Patrick Omameh to lucrative contracts, drafting a new left guard, and switching Flowers (a former first-round pick) to right tackle before ultimately discarding him, too.

The line has continued to struggle, but it was the beginning of a teamwide rebuilding effort, one that may take years to complete.

Shurmur continues to project confidence. The other day, he nodded to the Giants’ history of success.

“It’s in the Giants’ DNA to pull this thing out,” he said. “I’ve seen it, I’ve studied the history of it, and I like the locker room that we’ve assembled. I’m impressed with the guys that we have in the locker room, and I know they’re going to fight.”